The inherent contradictory nature of mathematics lies in the fact "axioms" are created in order to justify it leading to an inherent form of circularity where an axiom creates a framework, the framework is faulty hence a new axiom must be created.

http://gamahucherpress.yellowgum.com/wp ... MATICS.pdf

14

In Burali-fortis day there was a set of all ordinals which resulted in paradox This set has been outlawed in

set theory -because it sends it into self -contradiction. To avoid this paradox mathematicians ad hoc introduced the axiom called the Axiom schema of specification ie axiom of separation

http://en.wikipedia.org/wiki/Burali-Forti_paradox

“Modern axiomatic set theory such as ZF and ZFC circumvents this antinomy by simply not allowing construction of sets with unrestricted comprehension terms like "all sets with the property P",” Russell paradox In Russells day there was a set of all sets which destroyed naive set theory-sent it into contradiction-so to avoid it set theory just introduced an axiom Axiom schema of specification ie axiom of

separation Modern set theory just outlaws/blocks/bans this Russells paradox by the introduction of the ad hoc axiom the Axiom schema of specification ie axiom of separationwhich wiki says

http://en.wikipedia.org/wiki/Zermelo%E2 ... set_theory

15"The restriction to z is necessary to avoid Russell's paradox and its variants. " Thus we have two sets - which at one time did exist-which send maths into contradiction just being disallowed by adding an ad hoc axiom IT SHOULD BE NOTED THE IRONY HERE Russell created the axiom

of reducibility to to get rid of paradoxes in mathematics by outlawing impredicative statements but Zermelo created an ad hoc impredicative

axiom the axiom of separation to avoid many paradoxes ie Russell’s paradox Now there is double irony in this as many say Russells axiom of

reducibility should be outlawed as it is ad hoc but the same mathematicians will not say the axiom of separation should be outlawed or dropped as it is

ad hoc –HOW STRANGE Also the ad hoc creation of this impredicative axiom of separation means

1)ZFC is inconsistent

2) that the paradoxes it was meant to avoid are now still valid and thus mathematics is inconsistent As the axiom of ZFC ie axiom of separation outlaws/blocks/bans itself thus making ZFC inconsistent

Proof

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenk

el_set_theory

## The Contradictory Nature of ZFC Set Theory Rooted in Axiom Creation

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